# Auxiliary Library Documentation

By Jacob Neumann, June 2021

This page collects the documentation of the Auxiliary Library. The purpose of this library & its documentation is to serve as sample code, proofs, and worked examples for students learning functional programming and/or SML for the first time. Accordingly, this documentation also includes explicit mathematical definitions, proofs by induction, asymptotic analyses, and evaluation traces, as appropriate.

Unless otherwise stated, the documentation assumes SMLNJ v110.99. Some features (e.g. modifications of the pretty printer) might not work in other versions of SMLNJ.

All the documents & code here should be considered "work-in-progress". If you spot an error in either, you can report it here.

## Trees

Provides polymorphic binary trees in a structure Tree, with a couple basic methods for working with them.

In particular, this includes the functions inord and foldr, which are used in the OrdTreeSet functor (see Sets below) and critical to the associated representation independence result.

Code

## Timing

The Timing module includes types for encoding years, months, days, times, time zones, etc., as well as numerous utilities for working with them. This module primarily serves as an extended example of how to use custom SML datatypes to encode data, and how to take advantage of pattern matching to write elegant code. This module includes some imperative features (achieved utilizing basis modules unique to SMLNJ), such as stopwatches, countdown timers, and functions which obtain the current time.

Code

## Permute

The Permute module contains utilities for permutating and sorting lists. The functions in this module are polymorphic, and some of the sorting functions furthermore serve as examples of currying.

## CPS Iterate

The CPSIterate module allows for imperative-programming-esque loops, but defined entirely functionally and entirely in continuation passing style.

## Language

The Language module provides combinators for working with "languages": lists of values of some equality type Sigma. Connects to some of the classic theory of computation, as well as providing sufficient combinators to capture a fragment of the logic of regular expressions. Good showcase of Higher-Order Functions & Combinators.

## Regular Expressions

The Regexp module implements regular expressions in Standard ML. Parametrizes over an "alphabet" (equality) type Sigma, and implements a type ''Sigma regexp with a CPS/Exn-control-flow function match which performs regular expression matching. Includes a method for obtaining the language of a regular expression, implemented using the Language module (above).

Requires: Language.sml

## Sets

An implementation of sets in Standard ML. Includes EQ & ORD typeclasses, the SET signature, and three implementations: ListSet (sets are unordered, duplicate free lists), OrdListSet (sorted, duplicate-free lists), and OrdTreeSet (sorted, duplicate-free trees). The latter two are equivalent (as proven by a representation independence proof), but the superior time bounds of OrdTreeSet are not realized unless we can maintain a balance invariant -- providing motivation for red-black trees.

Requires: Tree.sml